18,807 research outputs found
An extension of disjunctive programming and its impact for compact tree formulations
In the 1970's, Balas introduced the concept of disjunctive programming, which
is optimization over unions of polyhedra. One main result of his theory is
that, given linear descriptions for each of the polyhedra to be taken in the
union, one can easily derive an extended formulation of the convex hull of the
union of these polyhedra. In this paper, we give a generalization of this
result by extending the polyhedral structure of the variables coupling the
polyhedra taken in the union. Using this generalized concept, we derive
polynomial size linear programming formulations (compact formulations) for a
well-known spanning tree approximation of Steiner trees, for Gomory-Hu trees,
and, as a consequence, of the minimum -cut problem (but not for the
associated -cut polyhedron). Recently, Kaibel and Loos (2010) introduced a
more involved framework called {\em polyhedral branching systems} to derive
extended formulations. The most parts of our model can be expressed in terms of
their framework. The value of our model can be seen in the fact that it
completes their framework by an interesting algorithmic aspect.Comment: 17 page
Volume of the polar of random sets and shadow systems
We obtain optimal inequalities for the volume of the polar of random sets,
generated for instance by the convex hull of independent random vectors in
Euclidean space. Extremizers are given by random vectors uniformly distributed
in Euclidean balls. This provides a random extension of the Blaschke-Santalo
inequality which, in turn, can be derived by the law of large numbers. The
method involves generalized shadow systems, their connection to Busemann type
inequalities, and how they interact with functional rearrangement inequalities
Robust capacitated trees and networks with uniform demands
We are interested in the design of robust (or resilient) capacitated rooted
Steiner networks in case of terminals with uniform demands. Formally, we are
given a graph, capacity and cost functions on the edges, a root, a subset of
nodes called terminals, and a bound k on the number of edge failures. We first
study the problem where k = 1 and the network that we want to design must be a
tree covering the root and the terminals: we give complexity results and
propose models to optimize both the cost of the tree and the number of
terminals disconnected from the root in the worst case of an edge failure,
while respecting the capacity constraints on the edges. Second, we consider the
problem of computing a minimum-cost survivable network, i.e., a network that
covers the root and terminals even after the removal of any k edges, while
still respecting the capacity constraints on the edges. We also consider the
possibility of protecting a given number of edges. We propose three different
formulations: a cut-set based formulation, a flow based one, and a bilevel one
(with an attacker and a defender). We propose algorithms to solve each
formulation and compare their efficiency
Optimal competitiveness for the Rectilinear Steiner Arborescence problem
We present optimal online algorithms for two related known problems involving
Steiner Arborescence, improving both the lower and the upper bounds. One of
them is the well studied continuous problem of the {\em Rectilinear Steiner
Arborescence} (). We improve the lower bound and the upper bound on the
competitive ratio for from and to
, where is the number of Steiner
points. This separates the competitive ratios of and the Symetric-,
two problems for which the bounds of Berman and Coulston is STOC 1997 were
identical. The second problem is one of the Multimedia Content Distribution
problems presented by Papadimitriou et al. in several papers and Charikar et
al. SODA 1998. It can be viewed as the discrete counterparts (or a network
counterpart) of . For this second problem we present tight bounds also in
terms of the network size, in addition to presenting tight bounds in terms of
the number of Steiner points (the latter are similar to those we derived for
)
Exact algorithms for the order picking problem
Order picking is the problem of collecting a set of products in a warehouse
in a minimum amount of time. It is currently a major bottleneck in supply-chain
because of its cost in time and labor force. This article presents two exact
and effective algorithms for this problem. Firstly, a sparse formulation in
mixed-integer programming is strengthened by preprocessing and valid
inequalities. Secondly, a dynamic programming approach generalizing known
algorithms for two or three cross-aisles is proposed and evaluated
experimentally. Performances of these algorithms are reported and compared with
the Traveling Salesman Problem (TSP) solver Concorde
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